We study Fano schemes mathrm{F}_k(X) for complete intersections X in a projective toric variety Ysubset mathbb {P}^n. Our strategy is to decompose mathrm{F}_k(X) into closed subschemes based on the irreducible decomposition of mathrm{F}_k(Y) as studied by Ilten and Zotine. We define the “expected dimension” for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in X when the expected dimension of mathrm{F}_k(X) is zero.